the other criteria that kicks some people out are the requirements for a K-12 "teaching certificate". if you don't get a degree in education, it can be difficult and time consuming to get a teaching certificate. a friend of mine some years back desperately wanted to teach high school math, and had a bachelor's in pure math, tutored for years and was well loved, but couldn't surmount the teaching certificate hurdles. so now he works in software QA.